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Theorem sepnsepolem2 48604
Description: Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 48605. Proof could be shortened by 1 step using ssdisjdr 48542. (Contributed by Zhi Wang, 1-Sep-2024.)
Hypothesis
Ref Expression
sepnsepolem2.1 (𝜑𝐽 ∈ Top)
Assertion
Ref Expression
sepnsepolem2 (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
Distinct variable groups:   𝑦,𝐷   𝑦,𝐽   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐷(𝑥)   𝐽(𝑥)

Proof of Theorem sepnsepolem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sepnsepolem2.1 . 2 (𝜑𝐽 ∈ Top)
2 id 22 . . 3 (𝐽 ∈ Top → 𝐽 ∈ Top)
3 sslin 4264 . . . . 5 (𝑧𝑦 → (𝑥𝑧) ⊆ (𝑥𝑦))
4 sseq0 4426 . . . . . 6 (((𝑥𝑧) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) = ∅) → (𝑥𝑧) = ∅)
54ex 412 . . . . 5 ((𝑥𝑧) ⊆ (𝑥𝑦) → ((𝑥𝑦) = ∅ → (𝑥𝑧) = ∅))
63, 5syl 17 . . . 4 (𝑧𝑦 → ((𝑥𝑦) = ∅ → (𝑥𝑧) = ∅))
76adantl 481 . . 3 ((𝐽 ∈ Top ∧ 𝑧𝑦) → ((𝑥𝑦) = ∅ → (𝑥𝑧) = ∅))
8 ineq2 4235 . . . . 5 (𝑦 = 𝑧 → (𝑥𝑦) = (𝑥𝑧))
98eqeq1d 2742 . . . 4 (𝑦 = 𝑧 → ((𝑥𝑦) = ∅ ↔ (𝑥𝑧) = ∅))
109adantl 481 . . 3 ((𝐽 ∈ Top ∧ 𝑦 = 𝑧) → ((𝑥𝑦) = ∅ ↔ (𝑥𝑧) = ∅))
112, 7, 10opnneieqv 48592 . 2 (𝐽 ∈ Top → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
121, 11syl 17 1 (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥𝑦) = ∅ ↔ ∃𝑦𝐽 (𝐷𝑦 ∧ (𝑥𝑦) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  wrex 3076  cin 3975  wss 3976  c0 4352  cfv 6575  Topctop 22922  neicnei 23128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-top 22923  df-nei 23129
This theorem is referenced by:  sepnsepo  48605
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